Condensed matter physics. Assignment 1

Assignment 1: Drude Theory of Metals

Poisson Distribution

1. In the Drude model the probability of an electron suffering a collision in any in-

finitesimal interval dt is just dt/τ .

a. Show that an electron picked at random at a given moment had no collision during the

preceding t seconds with probability e−t/τ . Show that it will have no collision during

the next t seconds with the same probability.

b. Show that the probability that the time interval between two successive collisions of an

electron falls in the range between t and t+ dt is (dt/τ)e−t/τ .

c. Show as a consequence of (a) that at any moment the mean time back to the last collision

(or up to the next collision) averaged over all electrons is τ .

d. Show as a consequence of (b) that the mean time between successive collisions of an

electron is τ .

e. Part (c) implies that at any moment the time T between the last and next collision

averaged over all electrons is 2τ . Explain why this is not inconsistent with the re-

sult in (d). (A thorough explanation should include a derivation of the probability

distribution for T ).

Joule Heating

2. Consider a metal at uniform temperature in a static uniform electric field E. An

electron experiences a collision, and then, after a time t, a second collision. In the Drude

model, energy is not conserved in collisions, for the mean speed of an electron emerging

from a collision does not depend on the energy that the electron acquired from the field

since the time of the preceding collision.

a. Show that the average energy lost to the ions in the second of two collisions separated

by a time t is (eEt)2/2m. (The average is over all directions in which the electron

emerged from the first collision.)

b. Show, using the result of Problem 1(b), that the average energy loss to the ions per

electron per collision is (eEτ)2/m, and hence that the average loss per cubic centimeter

Due date: Feb. 8, 2011, 11:00am 1

Condensed matter physics. Assignment 1

per second is (ne2τ/m)E2 = σE2. Deduce that the power loss in a wire of length L

and cross section A is I2R, where I is the current flowing and R is the resistance of

the wire.

Thomson Effect

3. Suppose that in addition to the applied electric field in Problem 2 there is also

a uniform temperature gradient ∇T in the metal. Since an electron emerges from a collision

at an energy determined by the local temperature, the energy lost in collisions will depend

on how far down the temperature gradient the electron travels between collisions, as well

as on how much energy it has gained from the electric field. Consequently the power lost

will contain a term proportional to E · ∇T (which is easily isolated from other terms since

it is the only term in the second-order energy loss that changes sign when the sign of

E is reversed). Show that this contribution is given in Drude model by a term of order

(neτ/m)(dε/dT )(E · ∇T ), where ε is the mean thermal energy per electron. (Calculate

the energy lost by a typical electron colliding at r, which made its last collision at r − d.

Assuming a fixed (that is, energy-independent) relaxation time τ , d can be found to linear

order in the field and temperature gradient by simple kinematic arguments, which is enough

to give the energy loss to second order.)

Helicon Waves

4. Suppose that a metal is placed in a uniform magnetic field H along the z-axis.

Let an AC electric field Ee−iωt be applied perpendicular to H.

a. If the electric field is circularly polarized (Ey = ±iEx) show that the equation,

j(ω) = σ(ω)E(ω),

must be generalized to,

jx =

(σ0

1− i(ω ∓ ωc)τ

)Ex, jy = ±ijx, jz = 0. (1)

Due date: Feb. 8, 2011, 11:00am 2

Condensed matter physics. Assignment 1

b. Show that, in conjunction with Eq (1), Maxwell’s equations Eq (1.31) book Aschroft,

have a solution

Ex = E0ei(kz−ωt), Ey = ±iEx, Ez = 0,

provided that k2c2 = εω2, where

ε(ω) = 1−ω2p

ω

(1

ω ∓ ωc + i/τ

).

c. Sketch ε(ω) for ω > 0 (choosing the polarization Ey = iEx) and demonstrate that

solution to k2c2 = εω2 exist for arbitrary k at frequencies ω > ωp and ω < ωc.

(Assume the high field condition ωcτ >> 1, and that even for hundreds of kilogauss,

ωp/ωc >> 1.)

d Show that when ω << ωc the relation between k and ω for the low-frequency solution is

ω = ωc

(k2c2

ω2p

).

This low- frequency wave, known as a helicon, has been observed in many metals.

Estimate the helicon frequency if the wavelength is 1 cm and the field is 10 kilogauss,

at typical metallic densities.

Surface Plasmons

5. An electromagnetic wave that can propagate along the surface of a metal compli-

cates the observation of ordinary (bulk) plasmons. Let the metal be contained in the half

space z > 0, z < 0 being vacuum. Assume that the electric charge density ρ appearing in

Maxwell’s equations vanishes both inside and outside the metal. (This dose not preclude a

surface charge density concentrated in the plane z = 0.) The surface plasmon is a solution

to Maxwell’s equations of the form:

Ex = Aeiqxe−kz, Ey = 0, Ez = Beiqxe−kz, z > 0; (2)

Ex = Ceiqxek′z, Ey = 0, Ez = Deiqxek

′z, z < 0; (3)

Due date: Feb. 8, 2011, 11:00am 3

Condensed matter physics. Assignment 1

where, q, K K ′ real, and K K ′ are positive.

a. Assuming the usual boundary conditions (E‖ continuous, (εE)⊥ continuous) and using

the Drude results (1.35) and (1.29) Book (Ascroft), find three equations relating q, K

and K ′ as functions of ω.

b. Assuming that ωτ >> 1, plot q2c2 as a function of ω2.

c. In the limit as qc >> ω, show that there is a solution at frequency ω = ωp/√

2. Show

from an examination of K and K ′ that the wave is confined to the surface. Describe

its polarization. This wave is known as a surface plasmon.

Due date: Feb. 8, 2011, 11:00am 4